In the scene when Gandalf tells everybody of what Pippin did the other night it goes to an overhead shot.This is where the slip-up came in.In the background,a lady walks by with a dog.If they had all those weird creatures back then,why would they have a dog?

Clearly you haven't watched or read the movie/books close enough. Middle-Earth does have weird creatures in it, but dogs can be there too! In the first movie there was one. Not to mention, in the books Farmer Maggot has dogs, and there are several dogs, cats, and horses throughout the movies and books.

Who says they can't have dogs? They have wolves. There are cats mentioned. They have perfectly normal animals in Middle-Earth. There were dogs mentioned in the book, Farmer Maggot had 3 of them. How on Middle-Earth is this a slip up?

If it was Peter Jackson's dog, it would have shown up in every other scene in all three movies and it would have been very obvious :o)
Also - no problem with the dog being there in the first place. There are LOTS of horses in this movie and horses are normal animals. Also in Bree in the first movie there is a cat and a ferret. Both normal. AND somewhere in the very beginning of FOTR:EE, it shows a hobbit milking a cow. Not all the Middle-Earth creatures are made-up.

This is clearly not a slip-up. One requirement is that you cannot think of a logical explanation for this. Unless your imagination needs fine-tuning, you must have skipped the requirements. Link, what were you thinking when you said how on Middle-Earth?

You mean because they have "creatures" they can't have animals we're familiar with too? So I guess there were no cats, rodents, birds etc. either? Off the top of my head the most obvious example of normal pets is in The Fellowship - one of the hobbits has a pet dog that barks at a wraiths whose asking for Shire/Baggins.

They also had horses in the movie, so how far fetched would having a dog be? Just because its a story that takes place in "Middle Earth" and they have many different creatures why does that mean that they can't have dogs?

This means your saying that horses shouldn't exist either, Farmer Maggot had dogs in the book and there was also a goat in the early scenes of FOTR. Nice try but I think I hear the bells of Wrong tower chiming. WRONG! WRONG!

They not only had pigs, horses, cows, and ponies, they also had "normal" flowers such as roses, and "normal" trees such as ash, oak, elm, and birch. A dog is actually not all that surprising, especially considering that Tolkien originally started work on The Silmarillion, the Hobbit, and The Lord of the Rings to write a believable mythology for his adopted homeland of England. Dogs were often companions, hunters, trackers, and protection for the English. The fact that the dog in question was a greyhound is the only factor that I am unsure about. The hobbit's dog in the Fellowship of the Ring was believable, bloodhounds were bred for tracking and could be trained to be good guard dogs, but I don't know if a greyhound would have been found in medieval England.
Love, K and J :)